Non-abelian number fields with very large class numbers

Maximal class numbers of CM number fields

“…Siegel obtained a proof of this result by establishing that h > c 1 ( )|d| h < c 2 (n)|d| 1 2 log |d| n−1 (1) for the class number h of any number field of discriminant d and degree n over Q. To show that the upper bound (1) is essentially sharp, Ankeny, Brauer and Chowla succeeded in proving what can be viewed as a generalization of Siegel's result.…”

Dihedral and cyclic extensions with large class numbers

“…Therefore, the Galois group of f (x, t) over Q(t) and Q(t) is either D 5 or C 5 . In order to distinguish it, we use the criterion in [13], page 42.…”

“…Hence for t = (6r + 3)(36r 2 + 36r + 18) with r ∈ S(X), the regulator R Kt (log d Kt ) 5 . Now we show that f (x, (6r + 3)(36r 2 + 36r + 18)) gives rise to a regular C 6 extension over Q(r).…”

“…We use the following result of Daileda [5] to obtain a bound for L(1, ρ): Let ρ be an l-dimensional complex representation of a Galois group. We assume L(s, ρ) is an entire Artin L-function and N is its conductor.…”

“…Then the discriminant of f (x, t) become (4t 3 + 28t 2 + 24t + 47) 2 . LetK t be the Galois closure of K t and Gal(K t /Q) D 5 . Then [23]), there exists a constant c f depending only on f such that for every prime q ≥ c f , there is an integer t q with q splits completely in K t for all t ≡ t q mod q.…”